On the topology of Diophantine approximation Spectra

TL;DR

This paper studies the structure of spectra of Diophantine approximation exponents, proving they are compact, connected, and semi-algebraic sets in certain cases, and describes spectra for specific exponent families.

Contribution

It establishes the topological properties of spectra of Diophantine exponents and characterizes their structure for the case n=3, including new descriptions for specific exponent families.

Findings

Spectra are compact and connected sets.

For n=3, spectra are semi-algebraic and closed under component-wise minimum.

Provides a description of spectra for a family of six exponents by Schmidt and Summerer.

Abstract

Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case $n=3$, we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For $n=3$, we also obtain a description of the spectrum of a family of six exponents recently introduced by Schmidt and Summerer.